Introduction

Multicomponent flows
consist of different chemical species
that are mixed at the molecular level and
generally share the same velocity
and temperature.
They differ from
multiphase flows
where the different phases are immiscible
(Drew and Passman 1999)
and only occupy a fraction of the total volume.
The chemical species
may also interact through chemical reactions
and the resulting
multicomponent reactive flows
are observed in various natural phenomena
and engineering applications.

Figure 1: Astrium-ESA reentry demonstrator Copyright Astrium-ESA.

In astronautics,
during reentry of a space ship
into Earth's atmosphere
as illustrated
with the Astrium-ESA reentry demonstrator
in Figure 1,
when the spaceship meets denser
parts of the atmosphere,
high temperatures arise behind
the detached bow
shock surrounding the vehicle.
Because of these high temperatures,
polyatomic gases
dissociate, species may ionize,
and dissociated molecules
may recombine at the
body of the spaceship.
A detailed knowledge of
the resulting multicomponent flow and
of the heat fluxes at the vehicle
body is
of fundamental importance
for a proper vehicle design
(Anderson 1989).

Combustion of
oil, coal or natural gas
is still the source of
more than 85% of
primary energy in the world and
a typical domestic
premixed laminar flame is presented in
Figure 2.
It is
of the greatest importance
to diminish fuel consumption as well as
the emission of pollutant
in power plants, aircraft engines
as well as car engines
(Williams 1985; Poinsot and Veynante 2005).
This notably requires understanding
cycle to cycle variation in piston engines,
combustion instabilities in industrial furnaces,
ignition and flash back in
aero gas turbines
and more generally
to understand flame structure and dynamics.

In chemical engineering,
chemical reactors may be of
various shapes
and are typically designed
to optimize a given set of chemical reactions.
The corresponding processes may be
highly complex with multiple reactant injections,
heating or cooling devices,
pumps to increase pressure,
homogeneous chemistry as well
as heterogeneous chemistry with catalysts
(Rosner 1986;
Kee et al. 2003;
Schmidt 2009).
Optimizing
reactors' shapes as well as chemical
processes again requires a detailed knowledge of
the corresponding multicomponent
reactive flows.

Last but not least,
the study of atmospheric pollution,
illustrated in Figure 3 with
a picture of pollution above the
Eiffel tower in Paris,
involves a myriad of trace reactive species.
These reactive species are notably responsible
for phenomena ranging
from urban photochemical smog,
acid deposition,
stratospheric ozone depletion,
to climate change
(Seinfeld and Pandis 2006).
Investigating multicomponent atmospheric
flows including the effect of aerosols and clouds
is of the highest importance
for the twenty-first century.

The fluxes and the production rates satisfy the mass conservation
relations
\(\sum_{k\in S} \fluxdiff_k=0\)
and
\(\sum_{k\in S} m_k \omega_k=0\)
and by summing the species equations
we recover the total mass conservation equation
\(\partial\rho/\partial t +
\dxb\scal(\rho \vitesse)=0\).
The species governing equations
(1)
have been written in terms
of the species
mass densities \(\rho_k\), \(k\in S\),
but equivalent formulations are easily
written as for instance in terms of
the species mass fractions
\(\massfraction_k=\rho_k/\rho\).
When the force acting on the species
reduces to gravity \(\force_k=\gravite\),
\(k\in S\),
the right members of
the momentum conservation equation
(2)
and of
the energy conservation equation
(3)
are simplified into
\(\rho\gravite\)
and
\(\rho\gravite\scal\vitesse\),
respectively.
These conservation equations
(1)-(3)
have to be completed by
the relations expressing
the thermodynamic properties like
\(p\) and \(e\),
the chemical production rates
\(\omega_k\), \(k\in S\),
and the
transport fluxes
\(\viscous\),
\(\fluxdiff_k\), \(k\in S\),
and \(\heatflux\)
defined in equations
(4)-(13).

Thermodynamics

In the framework of
ideal gas mixture
thermodynamics,
the pressure \(p\),
the internal energy per
unit mass \(e\)
and
the entropy per
unit mass \(s\)
may
be written
(Guggenheim 1962)
\[
\begin{equation}
\tag{4}
p = \sum_{k\in S} R T
\frac{\rho_k}{m_k},
\qquad
\rho e = \sum_{k\in S} \rho_k e_k(T),
\qquad
\rho s = \sum_{k\in S} \rho_k s_k(T,\rho_k),
\end{equation}
\]
where
\(R\) is the gas constant,
\(T\) the absolute temperature,
\(e_k\) the internal energy per unit mass of the \(k\)th species,
and
\(s_k\) the entropy per unit mass of the \(k\)th species.
The internal energy \(e_k\)
and entropy \(s_k\)
of the \(k\)th species are
given by
\[
\begin{equation}
\tag{5}
e_k
=
e_k^{\rm st} + \int_{T^{\rm st}}^T \!\! c_{vk}(T')\, dT',
\qquad
s_k
=
s_k^{\rm st}
+
\int_{T^{\rm st}}^T \!\!
\frac{c_{vk}(T^\prime)}{ T^\prime} \, dT^\prime
-
\frac{R }{ m_k}
\log
\frac{\rho_k \, R T^{\rm st}}{ m_k \, p^{\rm st} },
\qquad
k\in S,
\end{equation}
\]
where
\(e_k^{\rm st}\) is the
formation energy of the
\(k\)th species at the standard temperature \(T^{\rm st}\),
\(c_{vk}\) the constant volume specific heat of the \(k\)th species,
and
\(s_k^{\rm st}\) the formation entropy of the
\(k\)th species at the standard temperature
\(T^{\rm st}\) and standard pressure
\(p^{\rm st}\).
Introducing the mean molar weight \(m\) of the mixture,
defined by
\(\rho/m
=
\sum_{k\in S} \rho_k/m_k\),
the ideal gas state law
may also be written
\(p = \rho R T/m\).
Other thermodynamic functions
are
directly expressed in terms of
energy and entropy as for instance
the enthalpy
\(\rho h = \sum_{k\in S} \rho_k h_k(T)\)
and the Gibbs function
\(\rho g = \sum_{k\in S} \rho_k g_k(T,\rho_k)\)
with
\(h_k(T) = e_k(T) + R T/m_k\)
and
\(g_k(T,\rho_k) = h_k(T) - T s_k(T,\rho_k)\),
\(k\in S\).
Thermodynamic data required for each
species of the mixture
reduce to the temperature dependent
specific heat
\(c_{vk}(T)\)---often evaluated in polynomial form---and
the two integration constants
\(e_k^{\rm st}\) and
\(s_k^{\rm st}\)
that represent the
formation energy and entropy
of the \(k^{\rm th}\) species at the standard state.
The elemental composition
of the chemical species is also required
for evaluation the species mass as well as for
chemical equilibrium calculations
(Guggenheim 1962; Williams 1985).

Thermodynamics
of fluid systems are
classically introduced with the concept of
local state,
that is, the classical
laws of
thermostatics
are applied
locally and instantaneously
at any point in the fluid system
(de Groot and Mazur 1984).
More satisfactory
nonequilibrium
thermodynamics
are obtained
from molecular frameworks
like statistical mechanics or the kinetic theory of gases
and have a wider range
of validity
(de Groot and Mazur 1984;
Keizer 1987;
Giovangigli 1999).
The physical justification
of the existence of a local state
indeed arises from the Boltzmann equation
which shows that
the species distribution functions are essentially Maxwellian
distributions
when collisions are dominant.

Thermodynamics may further be generalized to encompass the
situation of nonideal fluids which are such that
the compressibility factor
\(\compres
=
pm/(\rho R T)\)
deviates from unity.
Nonideal thermodynamics are
especially important for supercritical fluids
and generally at high pressure
(Guggenheim 1962;
Giovangigli and Matuszewski 2012).
A typical example of nonideal thermodynamics
is that of a fluid
governed by Van der Waals equation of state
(Guggenheim 1962).

These rates may be obtained from the
mass action law or from
the kinetic
theory of dilute gases
when the chemical
characteristic times are
larger than
the mean free
times of the molecules and the characteristic times of
internal energy relaxation
(Giovangigli 1999;
Nagnibeda and Kustova 2009).
The reaction constants
\(\mathcal{K}_i^{\rm f}\) and
\(\mathcal{K}_i^{\rm b}\)
are functions of temperature and
are Maxwellian averaged
values of molecular chemical transition probabilities
and this implies
the reciprocity relations
\[
\begin{equation}
\tag{8}
{\mathcal K}_i^{\rm e}(T)
=
\frac{ {\mathcal K}_i^{\rm f} (T) }{ {\mathcal K}_i^{\rm b} (T)},
\qquad
\log {\mathcal K}_i^{\rm e}(T)
=
-
\sum_{k\in S}
(\nu_{ki}^{\rm b} - \nu_{ki}^{\rm f}) \frac{ m_k g_k(T, m_k) }{ R T},
\qquad
i\in{\mathfrak R},
\end{equation}
\]
where \(\mathcal{K}_i^{\rm e}(T)\)
is the equilibrium constant of the
\(i\)th reaction
(Giovangigli 1999).
The forward reaction constants
\({\mathcal K}_i^{\rm f}\),
\(i\in{\mathfrak R}\),
are usually evaluated with Arrhenius law
\begin{equation}
\tag{9}
{\mathcal K}_i^{\rm f}
=
\mathfrak{A}_i T^{\mathfrak{b}_i}
\exp \bigl( - \mathfrak{E}_i/R T \bigr),
\qquad
i\in{\mathfrak R},
\end{equation}
where
\(\mathfrak{A}_i\) is the preexponential factor,
\(\mathfrak{b}_i\)
the temperature exponent
and
\(\mathfrak{E}_i\)
the activation energy
of the \(i\)th reaction.
The data required for each chemical
reaction
then reduce to the stoichiometric coefficients
\(\nu_{ki}^{\rm f}\)
and
\(\nu_{ki}^{\rm b}\),
and the Arrhenius constants
\(\mathfrak{A}_i\),
\(\mathfrak{b}_i\),
and
\(\mathfrak{E}_i\),
assuming that the
species thermodynamic is known.
The chemical reaction
stoichiometric coefficients
\(\nu_{ki}^{\rm f}\)
and
\(\nu_{ki}^{\rm b}\)
are such that atomic elements are conserved.

The size of detailed chemical
reaction mechanisms
has been steadily increasing over the
past years ranging from a few species
and reactions to several thousand
of species interacting through tens
of thousands of chemical reactions
as for instance for
bio-fuel or
atmospheric pollution.

The law of mass action does not hold
for nonideal fluids
and
the proper form for
nonideal rates of progress has been obtained by
Marcelin (1910).
The nonideal rates may
directly be expressed in terms of
activities or chemical potentials.
There are also
perturbations of the chemical source terms due to the
perturbed species distribution functions
in the Navier-Stokes regime in the framework of the
kinetic theory of reacting gases
(Giovangigli 1999; Nagnibeda and Kustova 2009).

Transport fluxes

The transport fluxes
\(\viscous\), \(\fluxdiff_k\), \(k\in S\), and \(\heatflux\) due to macroscopic variable
gradients
may be obtained from
various macroscopic and molecular theories
(Waldman 1958;
Mori 1958;
Chapman and Cowling 1970;
Ferziger and Kaper 1972;
Keizer 1987;
Ern and Giovangigli 1994;
Giovangigli 1999;
Nagnibeda and Kustova 2009)
and are in the form
\[
\begin{align}
\tag{10}
\viscous = {}&
- \kappa (\dxb\scal\vitesse) \identite
- \eta
\bigl(
\dxb\vitesse
+
\dxb\vitesse^t
- \pdtiers
(\dxb\scal\vitesse) \identite
\bigr) ,
\\[3pt]
\tag{11}
\fluxdiff_k = {}&
-
\sum_{l \in S} \rho_k D_{kl} \forcediff_l
-
\rho Y_k \theta_k \dxb\log T,
\qquad k \in S,
\\[3pt]
\tag{12}
\heatflux = {}&
-
\widehat\lambda \,\dxb T
-
p \sum_{k \in S} \theta_k \forcediff_k
+
\sum_{k \in S} h_k \fluxdiff_k,
\end{align}
\]
where
\(\kappa\) denotes the bulk viscosity
(sometimes termed the volume viscosity),
\(\eta\) the shear viscosity,
\(\identite\) the three dimensional identity tensor,
\(D_{kl}\), \(k,l\in S\), the multicomponent diffusion coefficients,
\(\forcediff_k\), \(k\in S\), the species diffusion driving forces,
\(\theta_k\), \(k\in S\),
the species thermal diffusion coefficients,
\(\widehat\lambda\) the partial thermal conductivity,
and
\({}^t\) the transposition operator.
The mass fluxes may also be expressed in terms of the
species diffusion velocities
\(\vitdiff_k\),
\(k\in\eespe\),
defined by
\(\fluxdiff_k
=
\rho_k \vitdiff_k\),
\(k\in\eespe\).
The first term
in the expression
(10)
of the viscous tensor $\viscous$
represents a resistance to compression
and the second term a resistance to shear.
Incidentally, the bulk viscosity
\(\kappa\) is of the same order of magnitude than the
shear viscosity \(\eta\) for polyatomic gases
and its impact on
fast flows has been established
(Billet et al. 2008;
Bruno and Giovangigli 2011).
The first term in the expression
(11)
of the diffusion flux
\(\fluxdiff_k\)
yields diffusion effects due to mole
fraction gradients, pressure gradients, and
differences between specific forces
acting on the species.
The second term represents diffusion
arising from temperature gradients and is termed the Soret---or Ludwig Soret---effect.
The first term in the expression
(12)
of the heat flux $\heatflux$ represents
represents Fourier's law,
the second term corresponds to the Dufour effect, that is,
heat diffusion due to concentration gradients,
which is the analog of
the Soret effect,
and the third
represents
the transfer of energy due to species molecular diffusion.
The matrix of diffusion coefficients
\(D=(D_{kl})_{k,l\in\eespe}\)
is symmetric positive semi-definite
and the entropy production due
to diffusive processes
reads
\((p/T) \langle D\forcediff,\forcediff\rangle\)
with
\(\forcediff=
(\forcediff_1,\ldots,\forcediff_\nespe)^t\).
Letting
\(\massfraction =
(\massfraction_1,\ldots,\massfraction_\nespe)^t\)
where
\(\massfraction_k\) is the
mass fraction of the \(k\)th species,
\(\theta=
(\theta_1,\ldots,\theta_\nespe)^t\),
and
\(\langle,\rangle\) the
scalar product,
the diffusion matrix \(D\)
and the thermal diffusion
coefficients
\(\theta\)
satisfy the mass conservation constraints
\(D\massfraction=0\)
and
\(\langle\theta,\massfraction\rangle=0\)
guaranteeing
that
\(\sum_{k\in S} \fluxdiff_k=0\).
The species diffusion driving force
\(\forcediff_k\),
\(k\in\eespe\),
may be written
\[
\begin{equation}
\tag{13}
\forcediff_k =
\dxb \molefraction_k
+
( \molefraction_k-\massfraction_k ) \dxb \log p
+
\frac{\rho_k}{p}
(\force - \force_k),
\qquad k \in S,
\end{equation}
\]
where
\(\molefraction_k\),
\(k\in\eespe\),
denote the species mole fractions,
and
\(\force
=
\sum_{k\in\eespe}
\massfraction_k \force_k\) the averaged force.
When gravity is the
only force acting
on the mixture
the diffusion driving forces reduce to
\(\forcediff_k =
\dxb \molefraction_k
+ ( \molefraction_k-\massfraction_k ) \dxb \log p\).
One may equivalently use the
unconstrained diffusion driving forces
\(\widehat\forcediff_k =
(\dxb p_k - \rho_k \force_k)/p\),
\(k \in S\),
where
\(p_k\) denotes the partial pressure of the $k$th species,
since
\(\forcediff_k
=
\widehat\forcediff_k
-
\massfraction_k
(\dxb p
-
\rho \force)/p\).
Many equivalent
alternative formulation
may be derived for
multicomponent fluxes
as for instance in terms of
thermal diffusion ratios
but are
beyond the scope of
the present
short article
(Waldman 1958;
Chapman and Cowling 1970;
Ferziger and Kaper 1972;
Ern and Giovangigli 1994;
Giovangigli 1999).

Historically, the multicomponent fluxes
have first been written
from empirical laws prior to being
derived from the kinetic theory of gases
or
statistical mechanics.
Moreover, even if the structure
of multicomponent transport fluxes
may be derived empirically or
in the framework of
macroscopic theories,
only the kinetic theory of gases yield the
multicomponent transport coefficients.

Transport coefficients

The evaluation of the
transport coefficients
\(\kappa\),
\(\eta\),
\(\widehat\lambda\),
\(D=(D_{kl})_{k,l\in S}\),
and
\(\theta=(\theta_k)_{k\in S}\)
requires solving
linear systems derived from the
variational solution of systems of
Boltzmann linearized integral equations
(Waldman 1958;
Chapman and Cowling 1970;
Ferziger and Kaper 1972;
Ern and Giovangigli 1994).
The mathematical structure of the transport linear systems as well as
fast iterative algorithms for
evaluating the transport coefficients
have been obtained
(Ern and Giovangigli 1994;
Ern and Giovangigli 1996).
In practice,
for any
coefficient
\(\mu\),
the linear system takes on either
a regular form or a singular form
(Ern and Giovangigli 1994; Giovangigli 1999).
The singular form may be written
\[
\begin{equation}
\tag{14}
\left\{
\begin{array}{l}
G \alpha =
\beta,
\\[2pt]
\langle\tlsc, \alpha \rangle = 0,
\end{array}
\right.
\end{equation}
\]
where
the system matrix \(G\) is symmetric
positive semi-definite
with nullspace
spanned by a
vector
\(\tlsn\),
where \(\tlsc\) denotes
the constraint vector,
$\alpha$ and $\beta$
the unknown and right hand side vectors,
and the well posedness conditions
\(\langle\tlsn,\beta\rangle=0\)
and
\(\langle\tlsn,\tlsc\rangle\neq0\)
hold
(Ern and Giovangigli 1994).
The symmetry properties of the linear systems
and of the transport coefficients
are inherited from
the symmetry properties of the Boltzmann collision operator
(Waldmann 1958;
Ferziger and Kapper 1972;
Ern and Giovangigli 1994;
Giovangigli 1999).
The regular case is
simpler with
\(G\) symmetric positive definite
and without constraint
(Ern and Giovangigli 1994).
The
coefficient \(\mu\)
is then
obtained with
a scalar product
\(\mu = \langle\alpha,\beta'\rangle\).
Direct or iterative
numerical algorithms may be used to
solve the transport linear systems
but are out of the scope
of the present
article.
It is also possible to use
interpolation
empirical
expressions that
are typically in the form
\(\eta
=
\sum_{k\in S}
\molefraction_k \eta_k\)
where
\(\eta\) denotes the mixture viscosity,
\(\molefraction_k\) the mole fraction of the
\(k\)th species and
\(\eta_k\) the viscosity of the
\(k\)th species
(Ern and Giovangigli 1994).
Finally, there exists library of computer programs
which may be used for evaluating the
multicomponent transport coefficients
(Ern and Giovangigli (EGLIB)).

In order to illustrate
multicomponent diffusion,
the Stefan-Maxwell equations
associated with the species diffusion velocities
\(\vitdiff_k\),
\(k\in\eespe\)
are presented.
These
equations,
obtained at the leading order from the
kinetic theory of gases,
are in the form
\[
\begin{equation}
\tag{15}
\forcediff_k
=
\sum_{\doubleindices{l \in \eespe}{l \ne k}}
\frac{\molefraction_k \molefraction_l }{{\cal D}^\bin_{kl} }
\,\vitdiff_l
\,
-
\,
\sum_{\doubleindices{l \in \eespe}{l \ne k}}
\frac{\molefraction_k \molefraction_l }{ {\cal D}^\bin_{kl} }
\vitdiff_k,
\qquad
k \in \eespe,
\end{equation}
\]
where
\({\cal D}^\bin_{kl}(T,p)\) denotes the binary diffusion coefficient
of the species pair \((k,l)\).
These equations
must also be completed by
the constraint
\(\sum_{k \in \eespe} \massfraction_k \vitdiff_k = 0\)
associated with mass conservation.
An elementary derivation of
these equations has been given by Williams
(Williams 1958a).
The resulting expression for the
species diffusion velocities in terms of the
mole fraction gradients
appears to be complex
and couples all species.
This complex dependence
on concentration gradients
is illustrated
by the Duncan and Toor experiment
on ternary diffusion processes
(Duncan and Toor 1962)
where reverse diffusion has been observed in
full agreement with the Stefan-Maxwell equations.

Boundary conditions

The
description of
general
reactive
flow boundary conditions
may be found in the literature
(Oran and Boris 1987;
Kee et al. 2003).
Dirichlet boundary conditions are typically associated with
inflow phenomena in infinite length domains,
isothermal walls, or classical velocity adherence conditions.
Neumann boundary conditions are often
associated with symmetry boundaries,
adiabatic walls, or nonreactive walls.

When a gaseous mixture is in contact with a
solid body or a liquid layer,
the interfacial equations are
also the boundary conditions of the gas phase equations.
Typical interfacial equations
may involve conservation jump relations
for species mass, momentum and energy,
continuity of some variables
like temperature or tangential velocity,
heterogeneous surface chemistry
involving catalysts or solid species,
adherence conditions,
elastic
as well as thermal
interactions with solid structures.
The species boundary conditions at a reactive interface are
for instance in the form
\[
\begin{equation}
\tag{16}
\rho_k^{}
( \vitesse
+
\vitdiff_k^{} ) {\cdot} {\boldsymbol n}
=
m_k^{} \widehat\omega_k^{}, \qquad
k\in S,
\end{equation}
\]
where
\(\widehat\omega_k^{}\)
are the surface production rates.
These rates may take into account
catalysis,
film deposition or surface ablation
(Kee et al. 2003; Ern et al. 1996).
Interaction with boundaries
may also involve fluid-structure interaction,
evaporation,
triple points,
free boundaries,
and
radiative heat losses.

Simplified models

The complete system of
fundamental
equations
governing multicomponent reactive flows
presented
in the previous sections
may be used to model
various flows,
but, in a number of
situations, simplifications
may be introduced
following different ideas
(Giovangigli 1999).

A first idea is to simplify the reactive aspects of the
flow under consideration.
In this situation, the number of species and
chemical reactions
are decreased and the resulting set of partial differential equations is
simplified.
The transport fluxes and
transport property evaluation
may accordingly be simplified.
A typical example is that
of a single irreversible chemical reaction
(Williams 1985).
Another type of
chemistry
simplification is associated with the idea
of a slow manifold.
In this framework,
it is assumed that the state of the mixture,
after some fast relaxation process that may be discarded,
belongs to a manifold associated with a much slower dynamics.
The manifold
is then parametrized by
a small
set of parameters,
typically some concentrations or
thermal parameters,
that are governed by a reduced system of
partial differential equations.
In combustion science for instance,
slow manifolds have first been
defined by solely looking at the
source terms
(Peters 1985; Mass and Pope 1992)
and then defined through the
calculation of libraries of flamelets
thereby involving diffusive processes
(Gicquel et al. 2000;
Van Oijen et al. 2001;
Bykov and Maas 2007;
Auzillon et al. 2012).
The chemical equilibrium model may also be seen as an
ultimate simplified slow manifold model
where
the slow variables are
the atomic mass
densities,
momentum and energy.

A second idea is to simplify the fluid dynamics aspects of the problem.
This may be a geometrical simplification in
the problem, a similarity assumption
in the flow, or a simplification resulting from
an asymptotic limit.
As typical examples, we mention
continuously stirred reactors,
quasi one-dimensional flows,
creeping flows,
boundary layer flows,
viscous shock layer flows,
mixing layer flows,
inviscid flows,
or small Mach number flows.

Finally,
a third idea is to simplify the
coupling between chemistry and fluid dynamics.
However,
such a simplification is
only feasible
in very particular situations, since the coupling arises
through various terms in the complete equations.
Two typical situations are that of
an incompressible limit,
like the thermo-diffusive
approximation in flame theory,
or the dilution limit
where a
dilutant is in large concentration
and the reactive species are trace species.
Of course,
all ideas may also be used simultaneously,
so that the whole family
of resulting models is very large.

Mathematical structure and numerical methods

A convenient vector
notation is
introduced
in order to
recast
the multicomponent flow governing equations into
a compact form.
The mathematical structure of
multicomponent flow equations
is then addressed by
using symmetrized equations.
Such a structure is important for theoretical
as well as
numerical purposes.
Finally,
Computational reactive Fluid Dynamics---which
is nowadays a major tool in understanding of
complex flows---is discussed.

From the
symmetrized
form (18)
it is classically established
that the first order differential
operator
\(\widetilde A_0(\symev)
\partial/\partial_t
+
\sum_{i\in C}\widetilde A_i(\symev) \partial_i^{}\)
associated with convection
is hyperbolic whereas the
second order operator
\(\widetilde A_0(\symev)
\partial/\partial_t^{}
-
\sum_{i,j\in C}\widetilde B_{ij}(\symev) \partial_i^{}\partial_j^{}\)
associated with dissipative
phenomena is
degenerate parabolic.
Such a symmetric structure
is the consequence of the underlying
kinetic framework,
that is,
of symmetry properties
deduced from
the Boltzmann collision
operator
(Giovangigli 1999).
Moreover,
there is an important coupling
stability condition between the
hyperbolic and the parabolic operators,
the Kawashima-Shizuta condition
which physically states that all waves
associated with multicomponent Euler equations
are damped by dissipative processes
(Shizuta and Kawashima 1985;
Giovangigli and Massot 1998;
Giovangigli and Matuszewski 2013).
It is also possible to
split the variables between
hyperbolic and parabolic variables
(Kawashima and Shizuta 1988;
Giovangigli and Massot 1998).

The symmetrized forms may
notably
be used for
mathematical purposes
like existence theorems or
asymptotic stability results
(Vol'Pert and Hudjaev 1972;
Kawashima and Shizuta 1988;
Giovangigli 1999).
They may also be used for
finite element formulations
based on Streamline Upwind
Petrov-Galerkin techniques
(Hughes et al. 1986).

Computational reactive fluid dynamics

Computational Fluid Dynamics
is now a major tool in understanding of
complex flows
(Oran and Boris 1987;
Ferziger and Peric 1996;
Godlewski and Raviart 1996;
Laney 1998;
Chung 2002;
Anderson 2009;
Pletcher et al. 2013).
Numerical simulation of compressible flows is
a difficult task that
requires a solid background in fluid mechanics
and numerical analysis.
The nature of compressible flows may be very complex, with
features such as shock fronts, boundary layers, turbulence,
acoustic waves, or instabilities.

Taking into account chemical reactions
dramatically increases the difficulties,
especially when detailed chemical and transport models
are considered.
Interactions between chemistry and fluid mechanics are especially complex in
reentry problems
(Anderson 1989),
combustion phenomena
(Poinsot and Veynante 2005),
or chemical vapor
deposition reactors
(Hitchman and Jensen 1993;
Kee et al. 2003).
An important aspect of complex chemistry flows is
the presence of multiple time scales
which may range typically from \(10^{-10}\) second
up to several seconds.
In the presence of multiple time scales,
implicit methods are advantageous, since otherwise
explicit schemes
are limited by the smallest time scales
(Descombes and Massot 2004;
Oran and Boris 1987).
A second potential difficulty associated with the
multicomponent
aspect is the presence of multiple space scales.
In combustion applications for instance
the flame fronts are very thin and typically require
space steps of \(10^{-3}\) cm
at atmospheric pressure,
and even
\(10^{-5}\) cm
at \(100\) atm,
whereas a typical engine
scale may be of \(10\) cm
or even \(100\) cm.
The multiple scales can only be solved by using adaptive grids
obtained by successive refinements or by moving grids
for unsteady problems
(Smooke 1982;
Oran and Boris 1987;
Bennett and Smooke 1998;
Smooke 2013).
A goal of simplified models,
in addition to decreasing the number of
unknowns,
is also to suppress the fastest times scales
and the steepest gradients
in chemical fronts,
by eliminating also the most reactive intermediate species.

Nonlinear discrete equations may be solved
by using Newton's method or
any generalization (Smooke 1982; Smooke 2013).
The resulting large sparse
linear systems may then be solved by using
a Krylov-type method, such as GMRES.
Other sophisticated methods involve coupled
Newton-Krylov techniques
(Knoll et al. 1994),
time splitting algorithms
(Descombes and Massot 2004;
Nonaka et al. 2012),
higher order compact discretization schemes
(Noskov and Smooke 2005)
as well as
massively parallel simulations
(Chen 2011; Moureau et al. 2011).
Characteristic type boundary conditions
are often used for the simulation of reactive flows
(Poinsot and Veynante 2005).
Evaluating
aero-thermochemistry quantities is computationally expensive
since they involve multiple sums and products.
Optimal evaluation requires a low-level parallelization
depending on the problem granularity.
Moreover,
it is preferable, when writing numerical software,
to clearly separate the numerical tools
from the special type of equations that are under consideration.
In the context of multicomponent flows,
it is therefore a good idea to write
codes for general mixtures and
use libraries that automatically
evaluate thermochemistry properties
(Kee et al. 1980; Cantera)
and transport properties
(Ern and Giovangigli (EGLIB)).

Extended models

In the previous sections,
the fundamental modeling of multicomponent
flows,
the qualitative properties of the resulting
systems of partial differential equations,
and numerical methods
have been addressed.
In many practical situations,
however,
extended models are required
and some of these extensions
are briefly addressed in this section,
namely
turbulence modeling,
nonideal thermodynamics,
ionized flows,
thermodynamic nonequilibrium,
chemical equilibrium flows,
sprays,
and
radiation.
Non-Newtonian flows,
thin films,
biological flows,
relativistic flows,
or
quantum fluids
which
may all be multicomponent,
will not be addressed,
neither
heterogeneous multifluids---associated with multiphase flows---where
each phase may also be multicomponent
and
which are investigated elsewhere in Scholarpedia.

Turbulent flows

Turbulence
is one of the most complex phenomena in fluids
and turbulent flows are
encountered
in practical devices
like rockets,
aircraft engines,
industrial
furnaces,
chemical power plants
as well as
in the atmosphere.
Turbulence may be characterized by
fluctuations of all local flow properties
(Frisch 1995;
Lesieur et al. 2005;
Pope 2000;
Peters 2000;
Poinsot and Veynante 2005).
Turbulent flows may either be investigated by using
direct numerical simulation (DNS),
when all the physical scales are resolved,
or by using filtered equations
for Large Eddy simulations (LES)
or Reynolds-Averaged Navier-Stokes (RANS) simulations.

The LES or RANS equations
for turbulent flows are typically derived
by applying a filter or averaging operator, respectively,
to the set of
fundamental equations
presented in the previous sections
(Pope 2000; Peters 2000; Poinsot and Veynante 2005).
With LES the flow variables are filtered in the spectral space,
all frequencies greater than a given cut-off
are suppressed and
those lower than the cut-off are retained,
whereas with RANS
all flow quantities are averaged.
The unclosed correlations are then
expressed using
subgrid scale models
(Lesieur et al. 2005;
Pope 2000;
Poinsot and Veynante 2005).
Products of fluctuations are typically
modeled by gradient like laws
whereas the filtered chemical source term models
may involve wrinkled and strained fluctuating chemical fronts
as well as distributed reaction zones
depending on
the turbulence intensity
(Lesieur et al. 2005;
Pope 2000;
Poinsot and Veynante 2005).

Nonideal thermodynamics

Progress in the efficiency of automotive engines,
gas turbines and
rocket motors
have notably
been achieved with
high pressure combustion
(Candel et al. 2006).
As pressure is increasing,
attractive forces between molecules play a
more important role in fluids
and lead to
nonideal effects
so that the compressibility factor
\(\compres=pm/(\rho R T)\)
deviates from unity.
This is the case in particular above the
critical pressure where it is
possible to continuously change
a liquid like fluid into a gas like fluid
(Guggenheim 1962).

Nonideal multicomponent fluid thermodynamics are
often built from equations of state using
the compatibility with ideal gases as a limiting condition
(Guggenheim 1962; Giovangigli and Matuszewski 2012).
The chemistry sources are influenced by
nonidealities as well as
multicomponent diffusion which is then driven
by the gradient of chemical potentials
(Marcelin 1910;
Keizer 1987;
Giovangigli and Matuszewski 2012).
These nonidealities prevent
unphysical diffusion in
cold dense parts of the fluid.
The structure of the resulting set
of partial differential equations is further analyzed in
(Giovangigli and Matuszewski 2013).

Plasmas

Partially ionized gas mixtures
are related to a wide range of practical applications
including laboratory plasmas, high-speed gas
flows and atmospheric phenomena
(Braginskii 1958;
Chapman and Cowling 1970;
Ferziger and Kaper 1972;
Raizer 1987;
Bruno et al. 2003).
Another fundamental application is
inertial confinement fusion
where the thermonuclear fusion of light nuclei
is a source of energy
(Lindl 1998; Atzeni and Meyer-ter-Vehn 2009).
Application of the Chapman-Enskog method to partially
ionized gases is feasible for
low temperature high density plasmas
(Ferziger and Kaper 1972).
The interactions between particles at distances
greater than the Debye length
are considered to be mediated by the electric field
while those at shorter
distance are considered to be true collisions
(Ferziger and Kaper 1972).
We refer to
Raizer (1987),
Zhdanov (2002),
Nagnibeda and Kustova (2009),
Giovangigli and Graille (2009),
Graille et al. (2009),
Capitelli et al.(2012),
and
Capitelli et al.(2013)
for a detailed presentation of
the multicomponent plasmas governing equations.
In particular,
in strong magnetic fields,
the transport fluxes are
found to be
anisotropic
and different coefficients may be obtained depending
on the relative orientation
of variables gradients
with the magnetic field.
The corresponding macroscopic
equations have to
be completed by the
Maxwell equations governing the electric and magnetic fields.
Many simplifications are
also possible and
the physics of plasmas
is very rich and complex because of the many
characteristic lengths and times involved
(Ferziger and Kaper 1972; Raizer 1987).

Multitemperature flows

Thermodynamic nonequilibrium is
of fundamental importance in reentry problems,
laboratory and atmospheric plasmas, as well as discharges
or strong shock waves
(Zel'dovich and Raizer 2002;
Zhdanov 2002;
Capitelli et al. 2007;
Nagnibeda and Kustova 2009).
The most general
thermodynamic nonequilibrium
model is the
state to state
model where each internal state of
a molecule is independent and considered
as a separate species
(Capitelli et al. 2007;
Zhdanov 2002; Nagnibeda and Kustova 2009).
When there are partial equilibria between some of these
states, species
internal temperatures may
be defined
and the complexity of the model is correspondingly reduced
(Zhdanov 2002; Nagnibeda and Kustova 2009).
Another example is
that of electron temperature in plasmas
(Graille et al. 2009).
The next reduction step then
consists in equating some of the
species internal temperatures
and ultimately lead
to the one temperature flow
model presented in the previous sections
(Nagnibeda and Kustova 2009).

Chemical equilibrium flows

Chemical equilibrium flows are a limiting model
which is of interest for various applications
such as chemical vapor deposition reactors
(Gokoglu 1988),
flows around space vehicles
(Anderson 1989; Mottura et al. 1997),
or diverging nozzle rocket flows
(Williams 1985).
These simplified models
are valid when the characteristic
chemical times are small in comparison
with the flow time.
The equations governing chemical equilibrium flows
may either be derived
directly
in a kinetic framework
(Ern and Giovangigli 1998),
or by superimposing
chemical equilibrium
in the equations
presented in the previous sections.
Both methods lead to the same
conservations equations, transport fluxes, thermodynamics,
as well as
qualitative properties of transport coefficients
but yield different quantitative
values for the transport coefficients
(Ern and Giovangigli 1998).
The chemical equilibrium constraints
may then be used to eliminate the chemical
unknowns and
to reduce the model into a
system of partial differential equations
governing the slow variables that are
the atomic mass
densities,
momentum and energy
(Giovangigli 1999).
The chemical equilibrium model may also be seen as an
ultimately simplified slow manifold model.

Sprays and clouds

Many practical devices involve
dispersed
condensed phases
in the form of droplets or solid particles like
sprays, aerosols, mists,
dusts, clouds, fumes, suspensions,
or sooting flames.
Each of the condensed phase may
itself be multicomponent
and may interact with the multicomponent gas.
In these situations there are
often so many droplets or
solid particles that only a statistical
description is feasible through
the concept of distribution function
similar to that used in kinetic theory
(Williams 1958b; Williams 1985).
The corresponding Lagrangian models typically involve
Boltzmann type and
kinetic type spray equations
as introduced by
Williams (1958b, 1985).
The coupling between the
dispersed condensed phases and the
gas phase then arise through
vaporization,
condensation,
sublimation,
drag,
coalescence,
as well as
atomization
(Williams 1985).
The kinetic type equations may then
be discretized in a fully Lagrangian
way (O'Rourke 1985)
as well as in an Eulerian way
leading to multifluid models
(Laurent and Massot 1990; Fox et al. 2008).
When the condensed phases are not dispersed,
multiphase flows are obtained (Drew and Passman 1999)
and are discussed elsewhere in Scholarpedia.

Radiation

A radiant heat flux
may sometimes be added
to the heat flux
in the energy conservation equation
(Williams 1985; Zel'dovich and Raizer 2002).
This radiant heat flux is the integral of the radiant intensity
over all frequencies and all solid angles
and the
radiant intensity is governed by a
Boltzmann type equation
involving emission, absorption, and scattering coefficients
(Williams 1985).
Two classical approximated models in
radiation transport are
the optically thick or
optically thin media which lead---neglecting absorption
and scattering---to Stefan-Boltzmann type radiation
heat loss source terms
(Willimas 1985).
Radiant effects are also important at boundaries
which may absorb and emit radiant heat.

Examples of multicomponent flows

Three typical numerical
simulations
of multicomponent reactive flows
are presented in this section
in order to illustrate the preceding developments,
namely
a chemical vapor deposition reactor,
a direct numerical simulation of a
high pressure flame in a mixing layer
and
a reentry flow.

A chemical vapor deposition reactor

Chemical Vapor Deposition (CVD) is an industrially important process
used to produce solid films with extremely fine
compositional control and uniformity.
The influence of various operating parameters on
product quality
and on the chemical process in CVD reactors
may be investigated numerically.

Figures 4 illustrates the mole fraction isopleth
of
\({\rm Ga}({\rm C}{\rm H}_3)_3\)
and
\({\rm As}{\rm H}_3\)
in the
symmetry plane of the CVD reactor.
The inlet is on the left of the
symmetry plane and the outlet on the right.
At the bottom of the reactor
\(x=-1.5\)
the substrate
corresponds to the segment
\(z\in[4,5]\)
and the susceptor
to the segment
\(z\in[5,10]\).
Both reactive species
\({\rm Ga}({\rm C}{\rm H}_3)_3\)
and
\({\rm As}{\rm H}_3\)
are gradually
decomposed
with increasing temperature
above the heated substrate and susceptor
and are then carried outside the reactor.
Many intermediate species are formed that
interact chemically with
substrate and lead to crystal growth.
Figure 5 illustrates the mole fraction of
\({\rm As}{\rm H}_2\) which is formed by surface chemistry and
desorption,
of
\({\rm Ga}{\rm C}{\rm H}_3\)
which leads to carbon impurities in the crystal
as well as
\({\rm H}\) mainly present in the hot zone of the reactor.
In CVD systems
thermal diffusion
(Soret effet)
drives heavy reactant sources
away from the hot depletion zone
and plays a significant role in CVD modeling
(Ern et al. 1996).

A high pressure flame

A two-dimensional Hydrogen/Oxygen flame
stabilized behind a splitter plate
with a mean pressure
of 100 bar is investigated
(Ruiz et al. 2012).
At such high pressures,
above the critical pressure,
the fluids are nonideal,
and a real gas equation of state is used.
The
\({\rm O}_2\) fluid
is in a liquid-like dense state,
whereas the
\({\rm H}_2\) stream has
a gas-like density.
The two-dimensional splitter plate represents the lip of an
injector and the operating point is typical of a real engine.
The mixture involves the \(n = 8\) species
\({\rm H}_2\),
\({\rm O}_2\),
\({\rm H}_2{\rm O}\),
\({\rm H}\),
\({\rm O}\),
\({\rm O}{\rm H}\),
\({\rm H}{\rm O}_2\),
\({\rm H}_2{\rm O}_2\)
interacting through \(n^{\hskip-0.04em {\rm r}} = 12\) chemical reactions
(Ruiz et al. 2012).

Although turbulence is a 3D phenomenon,
the flame/flow interaction is mainly 2D in
the stabilization region and
the simplification to 2D is not a strong limitation.
Letting
\(h = 0.05\) cm
be
the splitter height,
the computational domain
is 11\(h\) long in the x-direction and 10\(h\) in
the y-direction.
Hydrogen is injected above the splitter at
a temperature \(T = 150\) K and a
velocity \(u = 125\) m/s.
Below the splitter, oxygen is fed at
\(T= 100\) K and \(u = 30\) m/s.
The shape of the inlet velocity profiles follows a 1/7th power law.
Although developed turbulence is generally present in the
feeding lines of rocket engines, no velocity perturbation
is added to the inflow boundary condition.
Yet, strong turbulence levels caused by vortex
shedding are observed downstream of the splitter
as illustrated in Figure 6
where the temperature field is presented,
allowing
for a developed turbulent mixing layer and strong
flame/turbulence interactions
(Ruiz et al. 2011).

The wall boundary conditions are that of radiative equilibrium with an emissivity
coefficient of \(0.8\), and
catalytic surface
reactions
at the wall
are not taken in account.
In Figure 7 are presented the Mach numbers around the
capsule with an angle of attack of 25 degrees
(Lani 2008).
The temperature behind the shock is
\(5000\) K and the pressure
\(10000\) Pa
to be compared with \(256\) K and \(35\) Pa
in front of the shock.